$A-B-C$ means that there is a function $f$ on the line containing $A,B,C$ such that $f(A)<f(B)<f(C)$. I think this is called the betweenness axiom in some geometry books.
My professor said that we need to show there is another function on the line containing $A,B,C$ that maps $A,B,C$ to a codomain such that $f(C) < f(B) < f(A)$. I'm kind of confused, can I just define that other function to be whatever I want? It seems like making the function $g(x) = -x$ would accomplish that. Am I allowed to just that function however I want?